Convolutional regularized least squares framework for reduced-order modeling of transonic flows

We develop a convolutional regularized least squares (CRLS) framework for reduced-order modeling of transonic flows with shocks. Conventional proper orthogonal decomposition (POD)-based reduced models are attractive because of their optimality and low online cost; however, they perform poorly when snapshots contain parameter-dependent discontinuities, leading to smeared shocks, stair-stepping, or nonphysical oscillations. In CRLS, we first map each full-order snapshot to a smoother representation by applying a one-dimensional Gaussian convolution with reflect padding along the flow field coordinates. The convolution hyperparameters (kernel width and support) are selected automatically by Bayesian optimization on a held out set of snapshots. POD bases are then extracted from the smoothed data, and the parametric dependence of the POD coefficients is learned via radial basis function interpolation. To recover sharp shock structures, we introduce an efficient deconvolution step formulated as a regularized least squares problem, where the regularization centers the reconstruction around a nearest-neighbor reference snapshot in parameter space. The resulting CRLS surrogate is evaluated on inviscid transonic flow over a transonic airfoil, modeled by the steady compressible Euler equations solved with the Stanford University unstructured code over a Latin hypercube sample of Mach number and angle of attack. Compared with standard POD and smoothed POD baselines, CRLS yields markedly improved shock location and strength, lower surface-pressure and field-level errors, and a 42% reduction in the number of POD modes required to capture a fixed fraction of snapshot energy. These results demonstrate that CRLS provides an accurate, data-efficient, and largely automated route to shock-aware reduced-order models for the high-speed aerodynamic design.